All Questions ⁺⁰237827 Questions

0

62

0

+1011

Algebra

Find the sum of all positive integers less than 1000 ending in 3 or 4 or 5.

gnistory Jun 30, 2024

0

49

1

+1011

Algebra

What is the coefficient of $x$ in $(x^3 + x^2 + x + 1)(x^4 - 8x^3 + 17x^2 - 23x + 14)$?

gnistory Jun 30, 2024

0

54

0

+1011

Algebra

There exists a polynomial $f(x)$ and a constant $k$ such that
(x^2 - 2x - 5) f(x) = 2x^4 + 19x^3 + kx^2 - 15x - 1.
What is $k?$

gnistory Jun 30, 2024

0

56

1

+1011

Algebra

Find $t$ if the expansion of the product of $x^3$ and $x^2 + tx$ has no $x^2$ term.

gnistory Jun 30, 2024

Jun 29, 2024

0

39

2

+64

halp

A magician makes potions by combining maple syrup from a magical maple tree with ordinary water. The magician starts with a large supply of two potions: a red potion, which is 60% magical syrup by volume (and the rest is just water), and blue potion, which read more ..

ghistory Jun 29, 2024

+1

54

1

+33

Geometry

In the diagram, ABCD is a square. Find Sin PAQ

jumbean Jun 29, 2024

-2

47

0

+400

Probability

(a) Let $ABC$ be an equilateral triangle, centered at $O.$ A point $P$ is chosen at random inside the triangle. Find the probability that $P$ is closer to $O$ than to any of the vertices. (In other words, find the probability that read more ..

Stanry Jun 29, 2024

0

60

1

+1524

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $x + y = 25$, then find the minimum value of $6x + 3y.$

learnmgcat Jun 29, 2024

0

46

1

+1524

Algebra

For certain values of k and m, the system
3a + 2b = 2
a + 2b = k - 8a - mb
has infinitely many solutions (a,b). What are k and m? read more ..

learnmgcat Jun 29, 2024

0

78

0

+1524

Algebra

The function $f(x)$ is defined only on domain $[-1,2]$, and is defined on this domain by the formula
$$f(x) = 2x^2-8x+1.$$
What is the range of $f(x)$? Express your answer as an interval or as a union of intervals.

learnmgcat Jun 29, 2024

0

36

1

+1524

algebra

Let
$$f(x) = \frac{1}{1+\frac{2}{1+\frac 3x}}.$$
There are three real numbers $x$ that are not in the domain of $f(x)$. What is the sum of those three numbers?

learnmgcat Jun 29, 2024

-2

75

0

+400

Number Theory

(a) Find the last two digits of $9*7*11$.
(b) Find the last two digits of $4^{5^6}$. (By convention, exponent towers are evaluated from the top down, so $4^{5^6} = 4^{(5^6)}$.)

Stanry Jun 29, 2024

-1

69

1

+400

Number Theory

(a) Find the last two digits of $9*7*8$.
(b) Find the last two digits of $2^{3^4}$. (By convention, exponent towers are evaluated from the top down, so $2^{3^4} = 2^{(3^4)}$.)

Stanry Jun 29, 2024

-1

67

0

+400

Number Theory

(a) Find the last two digits of $9*5*3$.
(b) Find the last two digits of $1^{2^3}$. (By convention, exponent towers are evaluated from the top down, so $1^{2^3} = 1^{(2^3)}$.)

Stanry Jun 29, 2024

-1

53

0

+400

Number Theory

Given
a &\equiv 1 \pmod{7} \\
b &\equiv 2 \pmod{7} \\
c &\equiv 6 \pmod{7} read more ..

Stanry Jun 29, 2024

-1

51

0

+400

Number Theory

Find the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by $2$.

Stanry Jun 29, 2024

-1

51

0

+400

Number Theory

Calculate $40*9 \pmod{23}.$ Express your answer as a non-negative integer that is less than $23$.

Stanry Jun 29, 2024

-1

56

0

+400

Number Theory

Find the remainder when $100^{100}$ is divided by $18$.

Stanry Jun 29, 2024

-1

73

0

+400

Number Theory

Find the last two digits of $16*{37}$.
(Compute the remainder when $16*{37}$ is divided by 100.)

Stanry Jun 29, 2024

-1

50

0

+400

Number Theory

Let $a$ be an integer such that $0 \le a \le 7$ and $a^2 \equiv a \pmod{8}$. If $a \neq 0,$ then find the value of $a$.

Stanry Jun 29, 2024

-1

65

0

+400

Number Theory

Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$. If $a \neq 0,$ then find the value of $a$.

Stanry Jun 29, 2024

-1

55

0

+400

Number Theory

Let $a$ be an integer such that $a \equiv 5 \pmod{7}$. Find the value of $a + 1 \pmod{7}$. Express your answer as a residue between $0$ and the modulus.

Stanry Jun 29, 2024

0

67

0

+900

Number Theory

Which of the residues $0,$ $1,$ $2,$ $3,$ $4,$ $5$ satisfy the congruence $x^2 \equiv 0 \pmod{6}?$
Give your answer as a list, separated by commas, in order from least to greatest.

eramsby1O1O Jun 29, 2024

0

61

0

+900

Number Theory

Which of the residues $0,$ $1,$ $2,$ $3,$ $4$ satisfy the congruence $3x \equiv 0 \pmod{5}?$
Give your answer as a list, separated by commas, in order from least to greatest.

eramsby1O1O Jun 29, 2024

0

74

0

+900

Number Theory

Which of the residues $0,$ $1,$ $2,$ $\dots,$ $11$ satisfy the congruence $3x \equiv 8 \pmod{5}?$
Give your answer as a list, separated by commas, in order from least to greatest.

eramsby1O1O Jun 29, 2024

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